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Full text of "Treatise On Analysis Vol-Ii"

360      XV    NORMED ALGEBRAS AND SPECTRAL THEORY

(15.8.1) Let b be a closed left ideal in A, and put Ub(x) - y = xy for all
y e b and all x e A. Then x\-> Ub(x) is a representation of the algebra A on the
Hilbert space b.

First of all, Ub(x) is a continuous operator on b, by (15.7.5.3). Clearly
Ub(xx') = Ub(x)Ub(xf). For y, z in b, we have

(Ub(x)*-y\z) = (y\Ub(x)  z) = (y\xz) = (x*y\z) = (Ub(x*)-y\z)

by virtue of (15.7.5.2), hence Ub(x)* = Ub(x*). Also if A has a unit element e,
then Ub(e) is the identity transformation.

When b = A, we write U(x) in place of UA(x); the representation U is
called the regular representation of A. It is faithful (1 5.5) by virtue of (1 5.7.5.7).
Moreover, x\-+ U(x) is a continuous linear mapping of A into =2?(A), by the
continuity of (x, y) i-* xy.

The study of complete Hilbert algebras is founded on the consideration of
their minimal left ideals and the idempotents which generate them.

For each left ideal I of A, we denote by I* the image of I under the involu-
tion ^H*,?*; clearly I* is a right ideal.

(15.8.2)    For each left ideal I of A, the orthogonal supplement I1 of the closure
I 0/1 is a left ideal.

Since I is a left ideal (15.1.3), we may as well assume that I is closed. If
y e I1 and z e A, then for each x e I we have (zy \ x) = (y \ z*x) = 0 because I is
a left ideal, whence it follows that zy e I1.

(15.8.3)    Lete^Q be an idempotent in A. Then

(i)   Ikll^i;

(ii)   e* is idempotent ;

(iii) the left ideal Ae is the set of all x e A such that x = xe, and is
closed in A.

The first assertion follows from the inequality ||i|| = \\e2\\ ^ \\e\\2. The
second is trivial. As to (iii), it is clear that if x = xe then x e Ae, and conversely
if x e Ae, then x = ye for some y e A, hence xe = ye2 = ye = x. The fact that
Ae is closed then follows from the continuity of the mapping xt~+x  xe,
and (3.1 5.1).

Consider in particular the self-adjoint idempotents (cf. (15.5.3.1 )):

(15.8.4)    Ifel9 e2 are self-adjoint idempotents in A, then the following proper-
ties are equivalent: (a) (e \e2) = 0; (b) ee2 == 0; (c) e2 et = 0.te Hilbert algebra.